= Le Cam lower bound under absolute-error loss
{c}
{title2=$\max_j\mathbb E_j|T-\theta_j|\geq\frac{\Delta}{2}(1-\operatorname{TV}(P_0,P_1))$}
For two observation laws $P_0,P_1$ at real parameters separated by $\Delta$, every <estimator> $T$ satisfies $\max_j\mathbb E_j|T-\theta_j|\geq\Delta(1-\operatorname{TV}(P_0,P_1))/2$. With common densities, the sum of the two risks is at least $\int\min(p_0,p_1)(|T-\theta_0|+|T-\theta_1|)$, at least $\Delta\int\min(p_0,p_1)=\Delta(1-\operatorname{TV})$ by the <triangle inequality>. Divide by two. The same proof works for any <metric> loss.
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